Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the boundary conditions for the equations of gas dynamics corresponding to the higher approximations in the Chapman-Enskog method of solution of the Boltzmann equation

  • 24 Accesses

  • 4 Citations

Abstract

When the gas-dynamic equations are derived from the solution of the Boltzmann equation by the Chapman-Enskog method, the order of the system of partial differential equations tends to increase with increasing number of the approximation. As a result, it is necessary to have more and more boundary conditions for these equations, which, however, are at present definitely known only for the first two approximations (models of an ideal gas and a viscous gas). A method is proposed for constructing additional boundary conditions; the method is illustrated in a number of examples.

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    S. Chapman and T. G. Cowling, Mathematical Theory of Non-Uniform Gases, Cambridge (1952).

  2. 2.

    M. N. Kogan, Dynamics of Rarefied Gases [in Russian], Nauka, Moscow (1967).

  3. 3.

    J. S. Darrozes, “Approximate solutions of the Boltzmann equation for flows past bodies of moderate curvature,” in: Rarefied Gas Dynamics, Vol. 1, Academic Press, New York (1969).

  4. 4.

    N. K. Makashev, “Solution of Botzmann's equation in problems of flow over bodies in the continuous-medium regime,” Tr. TsAGI, No. 1742 (1976).

  5. 5.

    V. N. Zhigulev, “On the equations of motion of a nonequilibrium medium with allowance for radiation,” Inzh. Zh.,4, No. 3 (1964).

  6. 6.

    V. S. Galkin, “Derivation of the equations of slow flows of gas mixtures from Boltzmann's equation,” TsAGI,5, No. 4 (1974).

  7. 7.

    V. S. Galkin, M. N. Kogan, and O. G. Fridlender, “On some kinetic effects in continuous-medium flows,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3 (1970).

  8. 8.

    V. S. Galkin, M. N. Kogan, and O. G. Fridlender, “On free convection in a gas in the absence of external forces,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3 (1971).

  9. 9.

    V. S. Galkin, M. N. Kogan, and O. G. Fridlender, “Concentration—stress convection and some properties of slow flows of gas mixtures,” Izv. Akad. Nauk SSSR, Mekh. Zhidk Gaza, No. 2 (1972).

  10. 10.

    V. A. Tsibarov, “Derivation of corrections to the principal moments of the distribution function of a viscous gas using an integral kinetic equation,” Vestn. LGU, Ser. Matem. Mekhan. i Astron., No. 7, Issue 2 (1966).

  11. 11.

    V. A. Tsibarov, “Chapman—Enskog distribution function and boundary conditions for the equation of motion,” in: Aerodynamics of Rarefied Gases, No. 6 [in Russian], State University, Leningrad (1973).

  12. 12.

    V. S. Galkin, M. N. Kogan, and O. G. Fridlender, “Thermal-stress and diffusion-stress phenomena,” in: Proc. Fourth All-Union Conf. on Rarefied Gas Dynamics and Molecular Gas Dynamics [in Russian], Izd. TsAGI, Moscow (1977).

  13. 13.

    M. Sh. Shavaliev, “Some results in the Burnett and super-Burnett approximations,” in: Proc. Fourth All-Union Conf. on Rarefied Gas Dynamics and Molecular Gas Dynamics [in Russian], Izd. TsAGI, Moscow (1977).

  14. 14.

    G. Grad, “Asymptotic theory of Boltzmann's equation,” in: Some Problems in the Kinetic Theory of Gases [Russian translations], Mir, Moscow (1965).

  15. 15.

    M. Sh. Shavaliev, “Burnett approximation to the distribution function and super-Burnett contributions to the stress tensor and heat flux,” Prikl. Mat. Mekh.,42, No. 4 (1978).

Download references

Author information

Additional information

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 77–87, May–June, 1979.

I should like to thank M. N. Kogan, V. S. Galkin, O. G. Fridlender, and S. A. Regirer for their interest in the work and for valuable discussions.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Makashev, N.K. On the boundary conditions for the equations of gas dynamics corresponding to the higher approximations in the Chapman-Enskog method of solution of the Boltzmann equation. Fluid Dyn 14, 385–394 (1979). https://doi.org/10.1007/BF01062444

Download citation

Keywords

  • Boundary Condition
  • Differential Equation
  • Partial Differential Equation
  • Boltzmann Equation
  • High Approximation